Determine whether the PDF converges in distribution

[cwbullivant]

Homework Statement

Let $$ \it{f}(x) $$ be a probability density function. Now let X_n have the density:

$$ \it{f}_{n}(x) = n\it{f}(nx) $$

Determine whether or not X_n converges in distribution to zero.

(this is the verbatim statement, there is no additional information given)

Homework Equations

[/B]
If the CDF $$ F_{n}(X) \rightarrow 0, n \rightarrow \infty $$ then X_n converges to zero in distribution.

The Attempt at a Solution

Definition of CDF:

$$ F_{n}(x) = P (X < x) = \int_{-\infty}^{x} \it{f}_{n}(x) dx= \int_{-\infty}^{x} n\it{f}(nx) dx $$

And to be a valid PDF, its integral from -∞ to +∞ must be 1. That requires that f(nx) must go to zero as nx goes to ±∞. So the lower limit on that integral is already taken care of; that part has to vanish if nf(nx) is going to be a valid PDF.

But for finite x, wouldn't the behavior of f(nx) as n goes to ∞ depend on exactly what f(nx) looks like (e.g. would go to zero if f(nx) = e^{-nx}, but not if f(nx) = ne^(-x))? I'm not sure on where to go from here.

 

[Ray Vickson]

Homework Statement

Let $$ \it{f}(x) $$ be a probability density function. Now let X_n have the density:

$$ \it{f}_{n}(x) = n\it{f}(nx) $$

Determine whether or not X_n converges in distribution to zero.

(this is the verbatim statement, there is no additional information given)

Homework Equations

[/B]
If the CDF $$ F_{n}(X) \rightarrow 0, n \rightarrow \infty $$ then X_n converges to zero in distribution.

The Attempt at a Solution

Definition of CDF:

$$ F_{n}(x) = P (X < x) = \int_{-\infty}^{x} \it{f}_{n}(x) dx= \int_{-\infty}^{x} n\it{f}(nx) dx $$

And to be a valid PDF, its integral from -∞ to +∞ must be 1. That requires that f(nx) must go to zero as nx goes to ±∞. So the lower limit on that integral is already taken care of; that part has to vanish if nf(nx) is going to be a valid PDF.

But for finite x, wouldn't the behavior of f(nx) as n goes to ∞ depend on exactly what f(nx) looks like (e.g. would go to zero if f(nx) = e^{-nx}, but not if f(nx) = ne^(-x))? I'm not sure on where to go from here.

Re-write $F_n(x)$ by changing variables to $u = nt$ in $\int_{-\infty}^{x} n f(nt)\, dt$. Note: do NOT $\int^x f(x) dx$; you should never, ever use the same symbol '$x$' to stand for two totally different things in the same expression---once as the integration variable, and once as the upper integration limit).

 

[Metmann]

Edit: Ok, I think I've written too much. First follow the hint of the post above.

 

[StoneTemplePython]

Definition of CDF:

$$ F_{n}(x) = P (X < x) = ... $$

This is wrong. A CDF is actually defined as $F(x) = P\big( X\leq x \big)$.

It's a technical but important distinction. A CDF is alway right continuous . This matters when you start dealing with CDFs involving discrete r.v.'s, hybrids, etc.

 

[Ray Vickson]

This is wrong. A CDF is actually defined as $F(x) = P\big( X\leq x \big)$.

It's a technical but important distinction. A CDF is alway right continuous . This matters when you start dealing with CDFs involving discrete r.v.'s, hybrids, etc.

Some books and papers (mostly old, now) actually define the CDF using "<". Probably 99.9% of sources nowadays use "≤", but I am not sure we can positively assert that "<" is wrong. (Maybe the OP's textbook and/or course notes take the "<" definition---unlikely, I know, but just barely possible.) However, I realize that many students are sloppy, and write "<" when they really mean "≤", so pointing it out to them is good!

 

[cwbullivant]

I realize that many students are sloppy, and write "<" when they really mean "≤", so pointing it out to them is good!

Sloppiness is exactly the issue here.

Anyhow, I make the appropriate substitution, with u = nt, du = n dt, and come out to:

$$ \int_{-infty}^{nx} \it{f}(u) du $$

This has eliminated the linear dependence on n from the original statement and reduced the problem to an integral with just a function in the integrand. From here, I don't seem to see much improvement. Whether or not the outcome of the integral goes to zero as n goes to infinity still seems like it should depend on the actual structure of f(u).

It was tempting to say that since the upper bound is u=nx, which is linear in n (which generally doesn't bode well for a finite answer as n goes to infinity), the PDF does not converge in distribution, but that doesn't sound right to me, because I don't see a good way to make that kind of conclusion from the integral of a completely general f(u).

 

[Metmann]

Whether or not the outcome of the integral goes to zero as n goes to infinity still seems like it should depend on the actual structure of f(u).

You can deduce the behavior of $f(y)$ for $y \rightarrow \infty$, namely $f$ should decrease faster than $\frac{1}{y}$.

 

[Ray Vickson]

Sloppiness is exactly the issue here.

Anyhow, I make the appropriate substitution, with u = nt, du = n dt, and come out to:

$$ \int_{-infty}^{nx} \it{f}(u) du $$

This has eliminated the linear dependence on n from the original statement and reduced the problem to an integral with just a function in the integrand. From here, I don't seem to see much improvement. Whether or not the outcome of the integral goes to zero as n goes to infinity still seems like it should depend on the actual structure of f(u).

It was tempting to say that since the upper bound is u=nx, which is linear in n (which generally doesn't bode well for a finite answer as n goes to infinity), the PDF does not converge in distribution, but that doesn't sound right to me, because I don't see a good way to make that kind of conclusion from the integral of a completely general f(u).

You are having trouble because you are using the wrong convergence criterion! Your stated criterion would be for $X_n \to +\infty$, not the $X_n \to 0$ posed in the question.

Recall the true definition: $X_n \to X$ in distribution if $F_n(x) \to F(x)$ for all $x$ (or, at least, for most $x$). What is the CDF $F(x)$ for $X = 0$ (the identically-zero random variable)?

 

[cwbullivant]

What is the CDF $F(x)$ for $X = 0$ (the identically-zero random variable)?

$F(x) = P(0 \leq x) = 1, x \geq 0, 0 otherwise$?